We receive the definition of P-AS convergence as follows:
On $(\Omega, \scr A,\rm P)$, a property holds true P-almost surely if $\exists$ set $N|[(P(N)=0) \forall \omega \in \Omega \text{ such that the result does not hold true}]$.
Unfortunately this definition is pretty semantic and thus I was wondering then if, for example, $N=\emptyset$ is a valid "null set" one could use in order to prove P-AS of some property.
Very rarely will $N=\emptyset$. If so, we say that the property holds $P$-surely.
The set $N$ will depend on the specific circumstance. For example, if you pick a number uniformly from $[0,1]$ then $P$-almost surely the number will be irrational. The $N$ in this case are the rational numbers in $[0,1]$.